Algebra switching – Field dividing an intermediate field

Consider an irreducible polynomial $ f (x) in F (x) $ in a polynomial ring of a field $ F $. Let $ L $ to be the dividing field of $ f (x) $ more than $ F $. Suppose that $ L $ is an extension of Galois on $ F $. Let $ alpha in L $ to be a root of $ f (x) $. Consider an intermediate field $ L-K-F $. Let $ g (x) in K (x) $ to be the minimal polynomial of $ alpha $. Let $ M $ to be a subfield of $ L $ which is the dividing field of $ g (x) $. Have we $ M = L $?

Assume that $ L = F (α) $ is a normal separable extension of F. Then, it follows that for any intermediate field $ L – K – F $, $ K (α) = L $. I'm trying to generalize this. Of course if $ f (x) $ in question is factored as $ g (x) h (x) $. It may be that in the area of ​​the split of $ g (x) $ more than K $, $ h (x) $ remains irreducible or at least not taken into account in linear polynomials. However, I do not find a counter-example.

If the answer to the question is negative, I have another question.

If the answer to the first question is negative, is there a plausible condition for having $ M = L $?

Thank you for your attention.